Free-by-cyclic groups are equationally Noetherian

TL;DR

This paper proves that all free-by-cyclic groups are equationally Noetherian, establishes properties of their growth rates, and analyzes their actions on trees, contributing to understanding their algebraic and geometric structures.

Contribution

It demonstrates that free-by-cyclic groups are equationally Noetherian and explores their actions on trees based on growth properties.

Findings

Free-by-cyclic groups are equationally Noetherian.

The set of exponential growth rates of these groups is well ordered.

Certain free-by-cyclic groups admit non-elementary 4-acylindrical actions on trees.

Abstract

A group $G$ is said to be equationally Noetherian if every system of equations in $G$ is equivalent to a finite subsystem. We show that all free-by-cyclic groups are equationally Noetherian. As a corollary, we deduce that the set of exponential growth rates of a free-by-cyclic group is well ordered. Along the way, we prove that free-by-cyclic groups with polynomially growing monodromies of infinite order admit non-elementary 4-acylindrical actions on trees. We show that the splittings arising from the improved relative train track machinery of Bestvina-Feighn-Handel are 2-acylindrical when the growth is at least quadratic.